The partial results of this study, which amounted to a number of quickly drawn diagrams, are shown below:

Study Overview

In 2006, in order to study and compare alpha male, beta male, gamma male, delta male, etc., and alpha female, beta female, etc., spatial distributions, density variations, and seating arrangements of humans, in confined situations, as compared to other animals, such as Norway rats, in experimental conditions, or turkeys, sage grouse, antelopes, gazelles, and bats, etc., in mating leks, as these two distributions relate or compare to arrangements of molecular species in the gas phase, the IoHT conducted a short survey in which a number of twenty-something Chicagoans, two of which were homecoming queens and one was alpha male, as based on the fact that he dated a homecoming queen, to draw detailed diagrams, from memory, of their high school cafeterias, showing the locations of each type of group, e.g. the in groups, the out groups, the most popular, etc., the age characterization of each group, i.e. freshman, sophomore, junior, and senior, as well as seating densities, and to draw the exact location of configurations such as tables, food-lines, walls, and particularly doors. Schools ranged between 800 to 6,500 students.

Disclaimer: to note, the following drawings and descriptions can be considered pejorative and derogatory with respect to the stereotypical terminology used. For the sake study trueness, however, each descriptor is labeled as per the terminology used by each study participant.

In addition, the probabilistic shape of any one human molecule is facilitated by using the the conception of human molecular orbitals (daily activity orbitals), as shown below:

In short, in regards to that of the spatial distribution or position of different types of molecules in any number of various systems, such as gas phase, liquid phase, surface phase, etc., molecular systems, different human and animal molecules have different sized molecular bubbles. Studies show, for example, that alpha males and alpha females are given more individual or personal space. A supermodel walking alone through a crowd of people will be given more personal space than as compared to a more homely female. [2]

When people are asked to approach a stranger, for example, and stop when they no longer feel comfortable, they will stop about two feet away from a tall person (22.7 inches to be exact) but less than a foot (9.8 inches) from a short person [1]; and as height is correlative with physical attractiveness, e.g. short men and women are less attractive than tall men and women, it is found that “very attractive people of any size are given bigger personal space and territory; which they carry around with them”. [2] In other words, “hot” molecules trigger volume increase be it a gaseous molecule or a human molecule. Similarly, in animal studies, in what is called the lek pecking order it is found that the more ‘physically-attractive’ leks, i.e. 'hot' leks, as defined via health, size, virility, opposite mate desirability criteria, etc., are given more spatial room in the center of the courtship and display assembly area, farthest away from the perimeter (exit or entry point), as shown below:

Moreover, the less attractive leks, i.e. 'cold' leks, are confined to the outer rims of the courtship circle. In an Uganda kob antelopes, for example, the bachelors are forced to roam the peripery of the lek, sometimes joining the nursery herd there but, their members are seldom if ever able to copulate. [3]

In these activity orbitals, humans are found moving about, probabilistically, within that of Edward T. Hall's 1966 conception of reaction bubbles (personal space), as shown below, in such a manner that only select individuals are allowed entry to different energy levels of a person's space: [1]

Interestingly, as the population densities began to build up, what Calhoun calls ‘behavioral sinks', aberrant behaviors such as disruptions in nest building, abnormal sexual courting rituals, increased levels of fighting, decreases in reproduction, increased levels of filth, social disorganization, food gathering inabilities, etc., began to accumulate in pens II and III.

Normally, for example, females work hard to keep litters sorted out and if a strange pup was introduced into the nest, the female would remove it. When nests were uncovered, the young would be moved to a new location that was more protected. Sink mothers, conversely, failed to sort out the young. Litters became mixed; the young were stepped on and often eaten by hyperactive males who invaded the nests. When a nest was exposed, the mothers would start moving the young but would fail to complete some phase of the move. Young carried outside the nest were often dropped and eaten by other rats. [1]

From a chemical point of view, molecular dynamics ensued such that pens I and IV each resulted in having a density of ten molecules, where as pens II and III each resulted in a density of fourteen molecules. Two alpha molecules first positioned into the end pens, farthest away from the boundary portholes; a number of alpha and beta females then attached to their spatial location; the remainder beta, gamma, and sigma molecules, etc., distributed in the remaining space according to more elaborate rules. Rats in pens II and III, for example, had to work harder. Three times daily there was a changing in the guard of the established hierarchy around the eating bins that was characterized by fighting and scuffling. Male sink rats, unable to establish stable territories, had to substitute time for space.

The dominant feature to note in this study is that alpha males tend to distribute themselves within the system in a distancing relationship to that of the boundaries. In bird leks, for example, which are breeding season mating grounds, territories tend to be circular, such that the perimeter of the circle functions as the boundary. A strict hierarchy in leks accords the most desirable, top-ranking, alpha males the most prestigious central territory, with ungraded and lesser aspirants ranged outside. Females attract to these arenas in due course to be fertilized, and normally they make their way through to one or other of the dominants in the centre. [6]

In high school cafeterias, a similar pattern develops. Specifically, herein, the IoHT conducted a short survey in which a number of twenty-something Chicagoans, two of which were homecoming queens and one was alpha male, as based on the fact that he dated a homecoming queen, to draw detailed diagrams, from memory, of their high school cafeterias, showing the locations of each type of group, e.g. the in groups, the out groups, the most popular, etc., the age characterization of each group, i.e. freshman, sophomore, junior, and senior, as well as seating densities, and to draw the exact location of configurations such as tables, food-lines, walls, and particularly doors. Schools ranged between 800 to 6,500 students. Meta-analysis of the results showed clearly that the senior alpha male tended to sit at a central location, farthest away from the doors (boundaries) of the cafeteria. According to a number of reports, as a sophomore, one would acquire status, year-by-year, the closer one sat to the senior table, and in particular in proximity to the alpha male. These results are shown below:

Discussion

Related to these population and seating density variations, is the conception of pressure, as defined in terms of atomic or molecular movements, and that of density, as defined in terms of atomic or molecular concentrations. The origin of these postulates derive from a theory developed by Swiss physicist and mathematician Daniel Bernoulli, as published in his 1738 Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.

In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. [7] Similarly, in 1873 Dutch physicist and chemist Johannes van der Waals published a treatise in which he outlined a revolutionary theory concerning the mutual attractions and interactions of molecules in the gas phase and how variations of these attractions result to create density, pressure, and temperature variations. Van der Waals, like Maxwell, had originally been inspired in this direction, i.e. towards the development of a theory of ‘intermolecular forces’, after he had read a popular 1857 treatise by Rudolf Clausius concerning the nature of the motion called heat. [8]

In his treatise, Clausius showed how Boyle’s law (1662), which states that the product of the pressure and volume of a fixed quantity of ideal gas, when held at a fixed temperature, is equal to a constant:

PV = k

can be readily derived on the assumption that a gas consists of material points which move at high velocity, that this velocity is of the order of that of sound and increases in proportion to the square root of the absolute temperature. This treatise was a revelation for van der Waals, because it then occurred to him that if a gas in the extremely dilute state, where the volume is so large that the molecules can be regarded as points, consists of small moving particles, which is obviously still so when the volume is reduced, then so to must be the case down to the maximum compression of the gas and also in liquids, which by this logic must be regarded as compressed gases at low temperature. [8]

By this line of reasoning, van der Waals conceived the idea that there is no essential difference between the gaseous and the liquid state of matter, and that the factors or forces which, apart from the motion of molecules, act to determine the pressure must be regarded as quantitatively different when the density changes and perhaps when the temperature changes, but that they must also be the same factors or forces that exercise their influence throughout the state of matter, be it a gas, liquid, or solid. Van der Waals visualized that molecules in the solid amorphous state impede each other’s motion owing to their close proximity. He also reasoned that the crystalline state has a molecular behavior and motion unique to itself.

On these views, van der Waals hypothesized that the reason why a non-dilute or non-ideal aggregate of moving particles fails to comply with Boyle’s law are firstly variations in the attraction between particles and secondly variations in their proper volume. Moreover, van der Waals reasoned that the point-like entities of the gas or liquid particles must not be considered material points but small particles with a real volume in common with all bodies known in nature. Herein, by comparison, we have shown that all molecules have numerous delineations of molecular bubbles confined to distinct molecular orbitals, which together define their volume.

Thus, to remedy the problem associated with the non-compliance of real gases and liquids with Boyle’s law, van der Waals conceived of the idea that there are variations in the mutual attractions between molecules, a factor now commonly defined as intermolecular forces, or weak yet defining social forces between molecues. [9] In his own words, in reference to these inter-molecular forces:

Since this force diminishes so quickly, according to van der Waals, it has two consequences, namely it results in the formation of molecular complexes, such as cliques, but that it also leads to the creation of a surface pressure. To elaborate on this, van der Waals states that "if we let the number of molecules that have combined into a complex be so large that it is possible to speak of a molecule at the center surrounded by a single layer containing almost as many other molecules as it is possible simultaneously, then for the surrounding molecules the attraction is directed towards the interior and acts to maintain the complex; and this part of its attraction is lost for the surface pressure."

The human or biological molecular picture we envision here is that of the alpha Norway rat surrounded by his catch of nine to ten females, in pens I and IV, or that of the common bird lek, circular in geometry, in which the most available males are found in the center, or that of the captain of the football team, during lunch break, sitting at the desired choice table of the most elite people in the school, surrounded, naturally enough, in near table proximity, by several dozen of the most available females in the school.

In the animal world, such as with peacocks, hammer-headed bats, or the New Zealand kakapo or owl parrot, etc., it is well known that females are attracted to lek groups more than they are to males all by themselves and that the larger the lek, the more pronounced the attraction. [6] In each of these groups, according to van der Waals, only the forces acting outward from these molecules can contribute to the formation of internal pressure, and that for both pseudo and true association, the number of complexes formed increases with decreasing temperature and volume.

Nearing the concluding point of his 1910 Nobel Lecture, on development of the equation of state for gases and liquids, van der Waals notes that it was of great importance for him to have been acquainted with Willard Gibbs’ 1876 treatises "on the equilibrium of heterogeneous substances", which Gibbs had sent him shortly after their appearance. The central concept gained from Gibbs’ treatises, according to van der Waals, was the principle that ‘for a given amount of substance, equilibrium sets in if the free energy is minimum for the temperature and volume.’ On this postulate, van der Waals introduced a parameter called the degree of association, which hypothetically exists between individual molecules, varying per relationship. Although he does not elaborate on this ‘degree of association’, he does point out that it is determined by means of Gibbs’ equation, and that the factors related to this degree of association must be introduced into the equation of state. [10]

An equation of state is simple a relation between state variables. [11] A state variable is any variable which represents the state of an object. In short, a state variable is a thermodynamic parameter that describes the momentary condition of a system, regardless of the path by which a system goes from one state to another, i.e. the sequence of intermediate states, such that the total change in any state variable will be the same. This means that the incremental changes in such variables are exact differentials. Examples of state variables include: density (ρ), energy (E), Helmholtz free energy (A), Gibbs free energy (G), enthalpy (H), internal energy (U), mass (m), pressure (P), entropy (S), temperature (T), and volume (V).

In conclusion of his lecture, van der Waals states that the connection between the Gibbs free energy equation, the degree of association between molecules, and the development of a state equation needed to exactly define the state of a molecular system exists, and suggests that ‘perhaps there is a direct way’. In this overview, we have outlined this path.

Moreover, according to the model outlined here, it is reasonable to postulate that both planetary formation as well as star formation may have coalesced in a similar manner. In other words, in a volume of empty space, it is reasonable to assume that first an α-molecule, or some variation thereof, attached, for whatever reason, to a certain local of space; thereafter, β-molecule, γ-molecule, and δ-molecules attracted likewise towards this center. In this progression, multitudes of attraction and repulsion tendencies working in coordination with volume or molecular bubble energy-regulatory delineations, would thus to form aggregates of larger size. As pressure increased centrally, the temperature would thus increase in a similar manner, such that hotter regions or hotter molecules would be found in the center. This is in accordance with the high-temperature molten core of the Earth as well as existence of hot thermonuclear reactions internal to the structure Sun. To summarize, it is likely that all mobile gas, liquid, and surface phase molecular systems arrange via an α-β-γ type distribution and that, in some way, these processes have bearing on the formation of larger molecular assemblies.

The molecular system interaction outline presented here was first enunciated in 1704 by the great English physicist and mathematician Isaac Newton, in the famous ‘Query 31’ to the Opticks, in which he states, quite accurately:

"Pseudo-association differs from true-association in that the latter is the result of new chemical forces which arise only when molecules are combined to form e.g. double molecules, whereas pseudo-association must be ascribed wholly to the normal molecular forces."

In 1910, in a lecture entitled ‘pseudo association’, given before the Royal Academy of Sciences in Amsterdam, van der Waals outlined a view in which molecules associate to form larger complexes, particularly in the liquid state, that the attraction of molecules decreases extremely quickly with distance, and that the attraction only has an appreciable value at distances close to the size of the molecule. In addition, van der Waals delineated between two types of forces that act between molecules:

"I have reduced this attraction, which acts in the whole volume, to a surface force which acts towards the interior and this, together with the external pressure, holds together the moving molecules."

In the Lek mating arena, shown above, which is modeled on the sage grouse, each alpha-male (highest ranking), beta-male, gamma-male, delta-male, etc., guards a territory of a few meters in size on average, and in which the dominate males may each attract up to eight or more females. [12] In addition, each individual is shown with variations in personal space (bubbles), where by higher-ranking individuals have larger personal space bubbles. [1] Common bird leks typically have 25-30 individuals.

In chimpanzee troops, the lower ranking beta, gamma, or omega males show a distancing respect to the alpha of the community by ritualized gestures such as bowing, allowing the alpha to walk first in a procession, or standing aside when the alpha challenges. [4] Canines also show deference to the alpha pair in their pack, by allowing them to be the first to eat and, usually, the only pair to mate; wolves are a good example of this. The status of the alpha is generally achieved by means of superior physical prowess. In certain highly social species, however, such as the bonobo, a contender can use more indirect methods, such as political alliances, enhanced intelligence, cleverness, or creativity, etc., to oust the ruling alpha and take his or her place.

As a result of these noted variations in the sizes of interpersonal bubbles, we might reasonably postulate that in all molecular systems there exist alpha-molecules, beta-molecules, gamma-molecules, delta-molecules, etc., ranked by hierarchy, being spatially distributed, that as a result of variations in intermolecular attractions, function to create macroscopic phenomena such as pressure or density variations in gas, liquid, or solid phase systems.

In this direction, rat population dynamics and interaction behaviors provide for an exemplary model of molecular distribution tendencies. Rats can be approximated as 23-element molecules. Beginning in 1947, ethologist John Calhoun conducted a fourteen-year research project in which he studied the population dynamics of wild Norway rats, through multiple generations, confined to quarter-acre outdoor pens surrounding a barn in Rockville, Maryland. [5] In the first stage of his study, Calhoun introduced five pregnant rats into the outdoor pen, being well-stocked with food and free from predators, and let observed the resulting reactions over a twenty-eight month period.

Interestingly, even though five pregnant over this time-span could theoretically produce 50,000 health progeny for this size pen, Calhoun found that the population never exceeded 200 individuals, and stabilized at 150, similar to the Dunbar number. Moreover, the rats were not randomly scattered throughout the pen area, but had organized themselves into twelve or thirteen local colonies of a dozen rats each. He noted that twelve rats is the maximum number that can live harmoniously in a natural group, beyond which stress and psychological effects function as group break-up forces.

Next, Calhoun ran six experiments, between 1958 and 1961, in which inside of the barn he built three 10 by 14-foot rooms open to observation through 3 by 5-foot glass windows cut into the floor of the hayloft. This arrangement permitted observers to have a complete view of the lighted room at anytime of the day or night without disturbing the rats. Each room was further divided into four pens by electrified partitions. Each pen was a complete dwelling unit, containing a food hopper, a drinking trough, places to nest (skyscraper type burrows for observation), and nesting materials. Ramp-like bridges passed over the specific electrified fences to connect pen I to pen II, pen II to pen III, and pen III to pen IV, thus creating a rectangular row of connected rat-cities or living units.

The results from the previous out-door study indicated that forty to forty-eight rats could healthily occupy each room. Thus, if the forty-eight rats were equally divided among four pens each pen would accommodate a colony of twelve rats, the maximum number a normal stable group can maintain before stress develops. Moreover, Calhoun reasoned that if he let the population build up to eighty, i.e. twice the stability number, the colonies would suffer a population collapse or die-off. To remedy this, Calhoun periodically removed rats so to then artificially observe behaviors typical to specific population densities, such as overpopulation.

In the normal uncrowned state, the young but physically mature male rats fight with each other until a fairly stable social hierarchy is established. In the first of two studies, two dominate alpha males established territories in pens I and IV. Each alpha male attracted a harem of eight to ten females. This type of arrangement was similarly balanced and consistent with the natural grouping among rats as observed in the quarter-acre outdoor experiment. The remaining fourteen male rats and associated females distributed themselves in pens II and III, as shown below:

Flamboyant dancing display by a male sage grouse performed in his defending spot within a compact area called a lek. The females, which are the smaller brown birds, congregate at the lek, and observe the prancing males before choosing one to mate with. [12]

"Have not the small Particles of Bodies certain Powers, Virtues, or Forces, by which they act at a distance … but also upon one another for producing a great Part of the Phenomena of Nature? For it’s well known, that Bodies act one upon another by the Attractions of Gravity, Magnetism, and Electricity; and these Instances shew the Tenor and Course of Nature, and make it not improbable by that there may be more attractive Powers than these. For Nature is very consonant and conformable to her self. How these Attractions may be perform’d, I do not here consider."